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Diagonal Harmonic Polynomials

Let $\mathbf{x}=x_1,x_2,\ldots,x_n$ and $\mathbf{y}=y_1,y_2,\ldots,y_n$ be two sets of $n$ variables. Diagonal harmonic polynomials are the solutions, in the polynomial ring $\mathbb{Q}[\mathbf{x},\mathbf{y}]$, of the system of partial differential equations

with one equation for each pair of integers $a$ and $b$, such that $a+b>0$. It has been shown by Haiman that the linear span of these polynomials has dimension $(n+1)^{n-1}$. The bigraded Frobenius characteristic of the resulting $\mathbb{S}_n$-module (under the diagonal action) is given by $\nabla(e_n)$ (See the $\nabla$-operator page.). It is also noteworthy that the bigraded enumeration of the alternating component of this module gives rise to the famous $q,t$-Catalan polynomials.

One may generalize to $k$ sets of variables. For this, one considers a $k\times n$ matrix $X=(x_{ij})$ of variables. The symmetric group $\mathbb{S}_n$ (of $n\times n$ permutation matrices) acts on these variables by multiplication on the right, whereas the general linear group $GL_k$ acts by multiplication on the left. The module of Diagonal Harmonic Polynomials is stable under both of the respective extensions of these action to polynomials in the variables $X$. Since the two actions commute, one may decompose the concerned module under joint irreducibles. This may be globally encoded in the “universal” format

There are higher versions of these modules, for all $m\in\mathbb{N}$, denoted by $\mathcal{D}_n^m$. When $k=1$ all these modules have dimension $n!$. For $k=2$, it has been established by Haiman that they have dimension $(mn+1)^{n-1}$; and I have conjecturedthat they have dimension $(m+1)^n(mn+1)^{n-2}$ when $k=3$. One also has universal expressions for the associated characters, such as